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The time to solve is generally a function of the number of vertices.
If there was an algorithm that could solve any instance of the problem in a time that was a polynomial function of the number of vertices, then there would be an algorithm to solve any NP-C problem in polynomial time.
Since no one knows how to do that, no one knows how to write a deterministic algorithm that can solve any instance of clique in polynomial time.
This does not prevent there from being an algorithm that can quickly solve a particular instance of clique.

But it doesn't really make sense to say that a particular problem can be solved in exponential time,
Exponential time and polynomial time describes how the time to solve varies as function of the size of the problem.

Any given instance of the problem can be solved in P time by an algorithm that knows the answer to that given instance.
But for any given algorithm there exist instances of the problem that cannot be solved in polynomial time, unless every other NP-Complete problem can also be solved in Polynomial time,

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